Three main stages may be marked in the development of the theory of the optical properties of metals. First, there is Drude's original theory, based on Maxwell's equations; in this theory the current density j at any point in a metal is supposed to be equal to the product of the electric vector of the light and of the conductivity of the metal. The theory yields the well-known Hagen-Rubens formula for the reflecting power, which appears to be in agreement with experiment for very long wave-lengths (λ > 10μ), but leads to completely incorrect results in the optical region. Various investigators ‡ have therefore modified the theory to take account of the finite mass of the electron; the formulae obtained pass over into the Drude formulae for sufficiently long wave-lengths. Finally a quantum theory of the phenomenon has been given by Kronig§, the electrons being treated as moving in a periodic field due to the crystal lattice in the manner originated by Bloch; this theory, in its turn, becomes identical with the modified classical theory if the periodic lattice is neglected.

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

Connected completely 0-simple semigroups are defined by a number of equivalent conditions, and a formula for the rank of these semigroups is proved. As a consequence an alternative proof of the result from [11] is given. In the case of a Rees matrix semigroup M0 [G, I, Λ, P] the rank is expressed in terms of |I|, |Λ|, G and a certain subgroup of G depending on P. At the end the minimal rank of all semigroups M0[G, I, Λ, P] is found for a given group G. Since every completely simple semigroup is connected, every result has a corollary for these semigroups.

Consider a random-walk problem on a simple lattice, the probabilities of the walker taking any direction in the lattice at each lattice point being equal. Then Polya (6) has shown that if a walker starts at the origin and continues to walk indefinitely, the probability of his passing through his starting point is unity in one and two dimensions, but less than unity in three or more dimensions. Recently, a generalization of this problem has been considered (1),(3) in which the walker is allowed to jump several lattice points with assigned probabilities. F. G. Foster and I. J. Good (3) have shown that if the assigned probabilities satisfy certain conditions, Polya's result still holds, and K. L. Chung and W. H. J. Fuchs (1) have shown that the result is valid under far less restrictive conditions. The above authors were primarily concerned with the question whether return is almost certain or not, and did not consider a detailed calculation of the probability at any stage. It is the purpose of the present paper to show that the use of contour integrals allied with the method of steepest descents (4) enables one to perform this calculation very simply.

In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their $N$-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of numbers for which the limiting frequencies of the digits in their $N$-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.

In this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group . These fields may be expanded as series of functions , where , m is fixed and the matrices Tl(g) form a 21 + 1 dimensional irreducible representation of .

Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with given l, n are transformed into others with the same values of l, n. That this must be so follows from Schur's Lemma and the fact that for each m and l the functions form a basis for an invariant subspace of functions on of dimension 2l + 1 in which an irreducible representation of acts. Explicit formulae for the results of such operations are presented.

The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).

This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.

1. Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (5), we find an expression (1) for the spectral radius r(A) of a finite square non-negative matrix A. This expression makes it very easy to study how r(A) behaves as a function of the diagonal elements of A.

We show that the seif-affine sets considered by McMullen [15] and Bedford [2] have infinite packing measure in their packing dimension θ except when all non-empty rows of the initial pattern have the same number of rectangles. More precisely, the packing measure is infinite in the gauge tθ|logt|−1 and zero in the gauge tθ|logt|−1−δ for any δ > 0.

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ[Bscr ] relative to [Bscr ], the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ[Bscr ] and a certain 1-design [Dscr ](B) induced by a block B ∈ [Bscr ]. The present paper studies the case where the size k of the blocks of [Dscr ](B) satisfies k = v − 1. In the general case, where k = v − 1 [ges ] 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on [Bscr ]. We prove that [Dscr ](B) contains no repeated blocks if and only if Γ[Bscr ] is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ[Bscr ] for some graph Γ with these properties. We prove further that Γ[B, C] ≅ Kv−1,v−1 if and only if Γ[Bscr ] is (G, 3)-arc transitive.

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averages

converge in norm.

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

We study functions $f$ of one complex variable with values in a complete locally convex space $E$. In [13] we had shown that for $f$ to be holomorphic it suffices that it is locally bounded and that there is a separating subset $H$ of $E'$ such that $x'\circ f$ is holomorphic for each functional $x'\in H$. We give here a new and considerably shorter proof of this result, and we extend it to a weak criterion for holomorphic continuability.

We derive some necessary conditions on a Riemannian metric (M, g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle over M. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.

We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface U is metrisable if and only if the induced (2, 2) conformal structure on M = TU admits a Kähler metric or a para–Kähler metric.

1. Introduction. In studying the elastic behaviour of inhomogeneous systems certain inclusion and inhomogeneity problems are fundamental. In the ‘transformation problem’, a region (the ‘inclusion’) of an unbounded homogeneous anisotropic elastic medium would undergo some prescribed infinitesimal uniform strain (because of some spontaneous change in its shape) if it were not for the constraint imposed by the surrounding matrix. When the inclusion has an ellipsoidal shape, Eshelby (3, 4) was able to show that the stress and strain fields within the constrained inclusion are uniform and that calculations could be completed when the medium was isotropic. A generally anisotropic medium seemed to raise forbidding analyses, but Eshelby (3) did point the way to an evaluation of the uniform strain which several authors (referred to later) developed into an expression amenable to numerical computation. Here we offer an elementary and immediate route to this expression of the uniform strain, which has been accessible hitherto only by the circuitous procedures of Fourier transforms. It is available as soon as the uniform state of strain in the inclusion is perceived and before an alternative evaluation is commenced. First, we appeal to a theorem (not it seems previously known) which reveals (in particular) the vanishing of the mean strain in the infinitesimally thin ellipsoidal homoeoid lying just outside the inclusion. Secondly, we need only reflect that at each point of the interface there is an immediate algebraic expression of the strain just outside the inclusion in terms of the uniform strain just inside.

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequence

where Xn is a projective module for − ∞ < n < + ∞, and

.

In order to model biological reactions, distances between knots/links based on tangle replacement are defined. Given a tangle R, an R-move is defined as the replacement of the zero tangle in a link L with the tangle R. The R-distance between the link L and the link M is defined to be the minimum number of R-moves required to change L into M where the minimum is taken over all diagrams of the link L. A formula is given to determine when one 4-plat knot/link can be obtained from another 4-plat via one R-move when R is a rational tangle and not equal to the 1/n tangle. The formula can also be used to find all rational tangles R for which the R-distance between two given 4-plats is 1 except in the case R = 1/n tangle. These results are also generalized to the case when any rational tangle P is replaced with any rational tangle R, P not necessarily the zero tangle.

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfying

for all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = K ∩ E is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull